In this work, we propose to explain the mass hierarchy and mixing pattern in the leptonic sector, we explore an extension of the Standard Model whose scalar sector includes one active and two inert doublets as well as some scalar singlets. The model includes a $S_3$ family symmetry supplemented by extra cyclic symmetries. As a consequence of our construction, a Dark Matter (DM) candidate is predicted and its properties are consistent with the observed cosmic abundances and the constraints imposed by direct and indirect detection experiments. The model allows Charged Lepton Flavor Violation (CLFV) processes like $\mu \rightarrow e \gamma $ and $\mu \rightarrow 3 e $, but the predicted branching ratios align with experimental limits. Additionally, our analysis elucidates the generation of the active neutrino masses through a one-loop radiative seesaw mechanism matching the observed neutrino oscillation data. The model agrees with experimental data on Higgs Diphoton decay rates and on oblique parameters.